Hippocrates practises and teaches medicine in about 400 BC on the Greek island of Kos. He will later be regarded as the father of medicine - partly because he is unlike his more theoretical contemporaries in paying close attention to the symptoms of disease, but also because a century or more after his death a group of medical works is gathered together under his name.

This Hippocratic Collection, and in particular the Hippocratic oath which is part of it, has remained the broad basis of medical principle up to our own day.

A slightly later Greek text, called On the Nature of Man and attributed to an author by the name of Polybus, introduces a medical theory which will be orthodox in Europe for some 2000 years. It states that human beings are composed of four substances or 'humours', just as inanimate matter is made up of four elements. India has a similar theory based on three.

The humours are blood, phlegm, black bile (melancholia) and yellow bile (chole). Too much of any one will give a person certain recognizable characteristics. He or she will be sanguine, phlegmatic, melancholy or choleric.

Aristotle's variable atoms: 4th century BC

Aristotle, practical as ever in his determination to get things worked out in detail, proposes a new theory to explain how the four elements of Empedocles and the atoms of Democritus produce the wide range of substances apprehended by our senses.

He suggests that there are two pairs of alternatives - hot and cold, moist and dry - which provide the exact nature of matter. In broad terms the four possible combinations are the four elements: earth (cold and dry), air (hot and moist), fire (hot and dry), water (cold and moist). But it is the infinitely variable balance between these qualities which creates the different atoms of stone or wood, bone or flesh.

Greek science in Alexandria: from the 3rd century BC

Classical Greece has produced a brilliant tradition of theorists, the dreamers of science. Attracted by the intellectual appeal of good theories, they are disinclined to engage in the manual labour of the laboratory where those theories might be tested.

This limitation is removed when the centre of the Greek world transfers, in the 3rd century BC, to Alexandria. In this bustling commercial centre, linked with long Egyptian traditions of skilled work in precious metals, people are interested in making practical use of Greek scientific theory. If Aristotle says that the difference in material substances is a matter of balance, then that balance might be changed. Copper might become gold.

Among the practical scientists of Alexandria are men who can be seen as the first alchemists and the first experimental chemists. Their trade, as workers in precious metals, involves melting gold and silver, mixing alloys, changing the colour of metals by mysterious process.

These are the activities of chemistry. The everyday items of a chemical laboratory - stills, furnaces, flasks - are all in use in Alexandria.

Euclid and Archimedes: 3rd century BC

Euclid teaches in Alexandria during the reign of Ptolemy. No details of his life are known, but his brilliance as a teacher is demonstrated in the Elements, his thirteen books of geometrical theorems. Many of the theorems derive from Euclid's predecessors (in particular Eudoxus), but Euclid presents them with a clarity which ensures the success of his work. It becomes Europe's standard textbook in geometry, retaining that position until the 19th century.

Archimedes is a student at Alexandria, possibly within the lifetime of Euclid. He returns to his native Syracuse, in Sicily, where he far exceeds the teacher in the originality of his geometrical researches.

The fame of Archimedes in history and legend derives largely from his practical inventions and discoveries, but he himself regards these as trivial compared to his work in pure geometry. He is most proud of his calculations of surface area and of volume in spheres and cylinders. He leaves the wish that his tomb be marked by a device of a sphere within a cylinder.

A selection of titles from his surviving treatises suggests well his range of interests: On the Sphere and the Cylinder; On Conoids and Spheroids; On Spirals; The Quadrature of the Parabola; or, closer to one of his practical discoveries, On Floating Bodies.

The earth and the sun: a heresy of the 3rd century BC

A lone voice on the Greek island of Samos. In about 270 BC Aristarchus is busy trying to work out the size of the sun and the moon and their distance from the earth. His only surviving work is on this topic, and his calculations are inevitably wide of the mark.

But references in other authors make it clear that his studies have brought him to a startling conclusion.

Aristarchus believes that the earth is in orbit round the sun (quite contrary to what is plain for anyone to see). There is an attempt, which comes to nothing, to have the man prosecuted for impiety. His idea joins the many other dotty notions which enliven the history of human thought, until Copernicus mentions him, in an early draft of his great book, as someone who had the right idea first.

On reflection Copernicus drops the name of Aristarchus from later versions of the text.

The circumference of the earth: calculated c. 220 BC

Eratosthenes, the librarian of the museum at Alexandria, has more on his mind than just looking after the scrolls. He is making a map of the stars (he will eventually catalogue nearly 700), and he is busy with his search for prime numbers; he does this by an infinitely laborious process now known as the Sieve of Eratosthenes.

But his most significant project is working out the circumference of the earth.

Eratosthenes hears that in noon at midsummer the sun shines straight down a well at Aswan, in the south of Egypt. He finds that on the same day of the year in Alexandria it casts a shadow 7.2 degrees from the vertical. If he can calculate the distance between Aswan and Alexandria, he will know the circumference of the earth (360 degrees instead of 7.2 degrees, or 50 times greater).

He discovers that camels take 50 days to make the journey from Aswan, and he measures an average day's walk by this fairly predictable beast of burden. It gives him a figure of about 46,000 km for the circumference of the earth. This is, amazingly, only 15% out (40,000 km is closer to the truth).